A comparison of noninvasive reconstruction of epicardial versus transmembrane potentials in consideration of the null space

Nonlocal based FISTA network for noninvasive cardiac transmembrane potential imaging

Objective. The primary aim of our study is to advance our understanding and diagnosis of cardiac diseases. We focus on the reconstruction of myocardial transmembrane potential (TMP) from body surface potential mapping.Approach. We introduce a novel methodology for the reconstruction of the dynamic distribution of TMP. This is achieved through the integration of convolutional neural networks with conventional optimization algorithms. Specifically, we utilize the subject-specific transfer matrix to describe the dynamic changes in TMP distribution and ECG observations at the body surface. To estimate the TMP distribution, we employ LNFISTA-Net, a learnable non-local regularized iterative shrinkage-thresholding network. The coupled estimation processes are iteratively repeated until convergence.Main results. Our experiments demonstrate the capabilities and benefits of this strategy. The results highlight the effectiveness of our approach in accurately estimating the TMP distribution, thereby providing a reliable method for the diagnosis of cardiac diseases.Significance. Our approach demonstrates promising results, highlighting its potential utility for a range of applications in the medical field. By providing a more accurate and dynamic reconstruction of TMP, our methodology could significantly improve the diagnosis and treatment of cardiac diseases, thereby contributing to advancements in healthcare.

An expert review of the inverse problem in electrocardiographic imaging for the non-invasive identification of atrial fibrillation drivers

BACKGROUND AND OBJECTIVE

Electrocardiographic imaging (ECGI) has emerged as a non-invasive approach to identify atrial fibrillation (AF) driver sources. This paper aims to collect and review the current research literature on the ECGI inverse problem, summarize the research progress, and propose potential research directions for the future.

METHODS AND RESULTS

The effectiveness and feasibility of using ECGI to map AF driver sources may be influenced by several factors, such as inaccuracies in the atrial model due to heart movement or deformation, noise interference in high-density body surface potential (BSP), inconvenient and time-consuming BSP acquisition, errors in solving the inverse problem, and incomplete interpretation of the AF driving source information derived from the reconstructed epicardial potential. We review the current research progress on these factors and discuss possible improvement directions. Additionally, we highlight the limitations of ECGI itself, including the lack of a gold standard to validate the accuracy of ECGI technology in locating AF drivers and the challenges associated with guiding AF ablation based on post-processed epicardial potentials due to the intrinsic difference between epicardial and endocardial potentials.

CONCLUSIONS

Before performing ablation, ECGI can provide operators with predictive information about the underlying locations of AF driver by non-invasively and globally mapping the biatrial electrical activity. In the future, endocardial catheter mapping technology may benefit from the use of ECGI to enhance the diagnosis and ablation of AF.

Reducing Line-of-block Artifacts in Cardiac Activation Maps Estimated Using ECG Imaging: A Comparison of Source Models and Estimation Methods

OBJECTIVE

To investigate cardiac activation maps estimated using electrocardiographic imaging and to find methods reducing line-of-block (LoB) artifacts, while preserving real LoBs.

METHODS

Body surface potentials were computed for 137 simulated ventricular excitations. Subsequently, the inverse problem was solved to obtain extracellular potentials (EP) and transmembrane voltages (TMV). From these, activation times (AT) were estimated using four methods and compared to the ground truth. This process was evaluated with two cardiac mesh resolutions. Factors contributing to LoB artifacts were identified by analyzing the impact of spatial and temporal smoothing on the morphology of source signals.

RESULTS

AT estimation using a spatiotemporal derivative performed better than using a temporal derivative. Compared to deflection-based AT estimation, correlation-based methods were less prone to LoB artifacts but performed worse in identifying real LoBs. Temporal smoothing could eliminate artifacts for TMVs but not for EPs, which could be linked to their temporal morphology. TMVs led to more accurate ATs on the septum than EPs. Mesh resolution had a negligible effect on inverse reconstructions, but small distances were important for cross-correlation-based estimation of AT delays.

CONCLUSION

LoB artifacts are mainly caused by the inherent spatial smoothing effect of the inverse reconstruction. Among the configurations evaluated, only deflection-based AT estimation in combination with TMVs and strong temporal smoothing can prevent LoB artifacts, while preserving real LoBs.

SIGNIFICANCE

Regions of slow conduction are of considerable clinical interest and LoB artifacts observed in non-invasive ATs can lead to misinterpretations. We addressed this problem by identifying factors causing such artifacts and methods to reduce them.

Noninvasive reconstruction of cardiac electrical activity: update on current methods, applications and challenges

Electrical activity at the level of the heart muscle can be noninvasively reconstructed from body-surface electrocardiograms (ECGs) and patient-specific torso-heart geometry. This modality, coined electrocardiographic imaging, could fill the gap between the noninvasive (low-resolution) 12-lead ECG and invasive (high-resolution) electrophysiology studies. Much progress has been made to establish electrocardiographic imaging, and clinical studies appear with increasing frequency. However, many assumptions and model choices are involved in its execution, and only limited validation has been performed. In this article, we will discuss the technical details, clinical applications and current limitations of commonly used methods in electrocardiographic imaging. It is important for clinicians to realise the influence of certain assumptions and model choices for correct and careful interpretation of the results. This, in combination with more extensive validation, will allow for exploitation of the full potential of noninvasive electrocardiographic imaging as a powerful clinical tool to expedite diagnosis, guide therapy and improve risk stratification.

Noninvasive identification of two lesions with local repolarization changes using two dipoles in inverse solution simulation study

BACKGROUND

The method for inverse localization and identification of two distinct simultaneous lesions with changed repolarization in the ventricular myocardium (two-vessel disease) is proposed and its robustness to errors in input data is tested in this simulation study.

METHOD

The inverse solution was obtained from the difference between STT integral body surface potential map computed with repolarization changes and the STT integral map from normal activation. In a numerical model of ventricles 48 cases of two simultaneous lesions and 48 cases of a single lesion were modeled. The effect of the lesions was taken to be represented by two dipoles. The input data were disturbed by three types of added noise. Twenty three characteristics of every obtained inverse solution were defined and four of them were used as the features in discriminant analysis task distinguishing the correct inverse solutions identifying two lesions.

RESULTS

The mean localization error for identified two lesions was 1.1±0.7cm. The sensitivity and specificity of quadratic discriminant analysis with cross-validation and feature selection was higher than 90%.

CONCLUSIONS

The combination of the inverse solution with two dipoles and discriminant analysis allows the identification of two simultaneous lesions without a priori information about the number of lesions.

Inverse Electrocardiographic Source Localization of Ischemia: An Optimization Framework and Finite Element Solution

With the goal of non-invasively localizing cardiac ischemic disease using body-surface potential recordings, we attempted to reconstruct the transmembrane potential (TMP) throughout the myocardium with the bidomain heart model. The task is an inverse source problem governed by partial differential equations (PDE). Our main contribution is solving the inverse problem within a PDE-constrained optimization framework that enables various physically-based constraints in both equality and inequality forms. We formulated the optimality conditions rigorously in the continuum before deriving finite element discretization, thereby making the optimization independent of discretization choice. Such a formulation was derived for the L₂-norm Tikhonov regularization and the total variation minimization. The subsequent numerical optimization was fulfilled by a primal-dual interior-point method tailored to our problem's specific structure. Our simulations used realistic, fiber-included heart models consisting of up to 18,000 nodes, much finer than any inverse models previously reported. With synthetic ischemia data we localized ischemic regions with roughly a 10% false-negative rate or a 20% false-positive rate under conditions up to 5% input noise. With ischemia data measured from animal experiments, we reconstructed TMPs with roughly 0.9 correlation with the ground truth. While precisely estimating the TMP in general cases remains an open problem, our study shows the feasibility of reconstructing TMP during the ST interval as a means of ischemia localization.

Computing ischemic regions in the heart with the bidomain model--first steps towards validation

We investigate whether it is possible to use the bidomain model and body surface potential maps (BSPMs) to compute the size and position of ischemic regions in the human heart. This leads to a severely ill posed inverse problem for a potential equation. We do not use the classical inverse problems of electrocardiography, in which the unknown sources are the epicardial potential distribution or the activation sequence. Instead we employ the bidomain theory to obtain a model that also enables identification of ischemic regions transmurally. This approach makes it possible to distinguish between subendocardial and transmural cases, only using the BSPM data. The main focus is on testing a previously published algorithm on clinical data, and the results are compared with images taken with perfusion scintigraphy. For the four patients involved in this study, the two modalities produce results that are rather similar: The relative differences between the center of mass and the size of the ischemic regions, suggested by the two modalities, are 10.8% ± 4.4% and 7.1% ± 4.6%, respectively. We also present some simulations which indicate that the methodology is robust with respect to uncertainties in important model parameters. However, in contrast to what has been observed in investigations only involving synthetic data, inequality constraints are needed to obtain sound results.

An electrical heart model incorporating real geometry and motion

This paper describes an electrical model of the ventricles incorporating real geometry and motion. Cardiac geometry and motion is obtained from segmentations of multiple-slice MRI time sequences. A static heart model developed previously is deformed to match the observed geometry using a novel shape registration algorithm. The resulting electrocardiograms and body surface potential maps are compared to a static simulation in the resting heart. These results demonstrate that introducing motion into the cardiac model modifies the ECG during the T wave at peak contraction of the ventricles.

Influence of individual torso geometry on inverse solution to 2 dipoles

BACKGROUND

The purpose of this study was to observe the influence of variety in individual torso geometries on the results of inverse solution to 2 dipoles.

METHODS

The inverse solution to 2 dipoles was computed from the measured data on 8 patients using either standard torso with various shapes and sizes of the heart and lungs in it or using various outer torso geometries with the same inhomogeneities. The vertical position of the heart relative to the fourth intercostal level was kept constant in all models. The results were compared with the reference solution computed in standard torso.

RESULTS

The inverse solution was influenced in 4 of 8 cases by changes of torso geometry and only in 1 of 8 cases by changes of internal inhomogeneities.

CONCLUSIONS

The use of individual torso geometry with the knowledge of the true heart position is very important for correct inverse results.

Application of L1-norm regularization to epicardial potential reconstruction based on gradient projection

The epicardial potential (EP)-targeted inverse problem of electrocardiography (ECG) has been widely investigated as it is demonstrated that EPs reflect underlying myocardial activity. It is a well-known ill-posed problem as small noises in input data may yield a highly unstable solution. Traditionally, L2-norm regularization methods have been proposed to solve this ill-posed problem. But the L2-norm penalty function inherently leads to considerable smoothing of the solution, which reduces the accuracy of distinguishing abnormalities and locating diseased regions. Directly using the L1-norm penalty function, however, may greatly increase computational complexity due to its non-differentiability. We propose an L1-norm regularization method in order to reduce the computational complexity and make rapid convergence possible. Variable splitting is employed to make the L1-norm penalty function differentiable based on the observation that both positive and negative potentials exist on the epicardial surface. Then, the inverse problem of ECG is further formulated as a bound-constrained quadratic problem, which can be efficiently solved by gradient projection in an iterative manner. Extensive experiments conducted on both synthetic data and real data demonstrate that the proposed method can handle both measurement noise and geometry noise and obtain more accurate results than previous L2- and L1-norm regularization methods, especially when the noises are large.

Application of kernel principal component analysis and support vector regression for reconstruction of cardiac transmembrane potentials

Non-invasively reconstructing the transmembrane potentials (TMPs) from body surface potentials (BSPs) constitutes one form of the inverse ECG problem that can be treated as a regression problem with multi-inputs and multi-outputs, and which can be solved using the support vector regression (SVR) method. In developing an effective SVR model, feature extraction is an important task for pre-processing the original input data. This paper proposes the application of principal component analysis (PCA) and kernel principal component analysis (KPCA) to the SVR method for feature extraction. Also, the genetic algorithm and simplex optimization method is invoked to determine the hyper-parameters of the SVR. Based on the realistic heart-torso model, the equivalent double-layer source method is applied to generate the data set for training and testing the SVR model. The experimental results show that the SVR method with feature extraction (PCA-SVR and KPCA-SVR) can perform better than that without the extract feature extraction (single SVR) in terms of the reconstruction of the TMPs on epi- and endocardial surfaces. Moreover, compared with the PCA-SVR, the KPCA-SVR features good approximation and generalization ability when reconstructing the TMPs.

Finite-element-based discretization and regularization strategies for 3-D inverse electrocardiography

We consider the inverse electrocardiographic problem of computing epicardial potentials from a body-surface potential map. We study how to improve numerical approximation of the inverse problem when the finite-element method is used. Being ill-posed, the inverse problem requires different discretization strategies from its corresponding forward problem. We propose refinement guidelines that specifically address the ill-posedness of the problem. The resulting guidelines necessitate the use of hybrid finite elements composed of tetrahedra and prism elements. Also, in order to maintain consistent numerical quality when the inverse problem is discretized into different scales, we propose a new family of regularizers using the variational principle underlying finite-element methods. These variational-formed regularizers serve as an alternative to the traditional Tikhonov regularizers, but preserves the L(2) norm and thereby achieves consistent regularization in multiscale simulations. The variational formulation also enables a simple construction of the discrete gradient operator over irregular meshes, which is difficult to define in traditional discretization schemes. We validated our hybrid element technique and the variational regularizers by simulations on a realistic 3-D torso/heart model with empirical heart data. Results show that discretization based on our proposed strategies mitigates the ill-conditioning and improves the inverse solution, and that the variational formulation may benefit a broader range of potential-based bioelectric problems.

A computationally efficient method for determining the size and location of myocardial ischemia

The purpose of this paper is to introduce a new method for solving the inverse problem of locating ischemic regions in the heart. The electrical activity in the human heart is modeled by the bidomain equations, which can be expanded to compute the potentials on the body surface. The associated inverse problem is to use ECG recordings to gain information about ischemias. We propose an algorithm for doing this, combining the level set method with a simpler minimization problem. Instead of trying to determine the shape, as in the level set method, we simply make the approximation that the ischemia is spherical. The effects of ischemia on the electrical attributes of heart tissue are examined. The new method is tested with computer simulations on synthetic body surface potential maps (BSPMs) in a realistic geometry, using realistic values for the parameters. It is shown to be, in some respects, superior to the level set approach and may be a step toward a practical algorithm useful in medical diagnostics.

Resolution strategies for the finite-element-based solution of the ECG inverse problem

Successful employment of numerical techniques for the solution of forward and inverse ECG problems requires the ability to both quantify and minimize approximation errors introduced as part of the discretization process. Our objective is to develop discretization and refinement strategies involving hybrid-shaped finite elements so as to minimize approximation errors for the ECG inverse problem. We examine both the ill-posedness of the mathematical inverse problem and the ill-conditioning of the discretized system in order to propose strategies specifically designed for the ECG inverse problem. We demonstrate that previous discretization and approximation strategies may worsen the properties of the inverse problem approximation. We then demonstrate the efficacy of our strategies on both a simplified and a realistic 2-D torso model.

Towards a computational method for imaging the extracellular potassium concentration during regional ischemia

We investigate the possibility of using body surface potential maps to image the extracellular potassium concentration during regional ischemia. The problem is formulated as an inverse problem based on a linear approximation of the bidomain model, where we minimize the difference between the results of the model and observations of body surface potentials. The minimization problem is solved by a one-shot technique, where the original PDE system, an adjoint problem, and the relation describing the minimum, are solved simultaneously. This formulation of the problem requires the solution of a 5 x 5 system of linear partial differential equations. The performance of the model is investigated by performing tests based on synthetic data. We find that the model will in many cases detect the correct position and approximate size of the ischemic regions, while some cases are more difficult to locate. It is observed that a simple post-processing of the results produces images that are qualitatively very similar to the true solution.

Truncated total least squares: a new regularization method for the solution of ECG inverse problems

The reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem which generally requires a regularized numerical solution. Two kinds of errors/noise: geometric errors and measurement errors exist in the ECG inverse problem and make the solution of such problem more difficulty. In particular, geometric errors will directly affect the calculation of transfer matrix A in the linear system equation AX = B. In this paper, we have applied the truncated total least squares (TTLS) method to reconstruct EPs from BSPs. This method accounts for the noise/errors on both sides of the system equation and treats geometric errors in a new fashion. The algorithm is tested using a realistically shaped heart-lung-torso model with inhomogeneous conductivities. The h-adaptive boundary element method [h-BEM, a BEM mesh adaptation scheme which starts from preset meshes and then refines (adds/removes) grid with fixed order of interpolation function and prescribed numerical accuracy] is used for the forward modeling and the TTLS is applied for inverse solutions and its performance is also compared with conventional regularization approaches such as Tikhonov and truncated single value decomposition (TSVD) with zeroth-, first-, and second-order. The simulation results demonstrate that TTLS can obtain similar results in the situation of measurement noise only but performs better than Tikhonov and TSVD methods where geometric errors are involved, and that the zeroth-order regularization is the optimal choice for the ECG inverse problem. This investigation suggests that TTLS is able to robustly reconstruct EPs from BSPs and is a promising alternative method for the solution of ECG inverse problems.

On the possibility for computing the transmembrane potential in the heart with a one shot method: an inverse problem

We analyze the possibility for using body surface potential maps (BSPMs), a priori information about the voltage distribution in the heart and the bidomain equations to compute the transmembrane potential throughout the myocardium. Our approach is defined in terms of an inverse problem for elliptic partial differential equations (PDEs). More precisely, we formulate it in terms of an output least squares framework in which a goal functional is minimized subject to suitable PDE constraints. The problem is highly unstable and, even under optimal recording conditions, it does not have a unique solution. We propose a methodology for stabilizing and enforcing uniqueness for this inverse problem. Moreover, a fully implicit method for solving the involved minimization problem is presented. In other words, we show how one may solve it in terms of a system consisting of three linear elliptic PDEs, i.e. we derive a so-called one shot method (also commonly referred to as an all-at-once method). Finally, our theoretical findings are illuminated by a series of numerical experiments. These examples indicate that, in the presence of regional ischemia, it might be possible to approximately recover the transmembrane potential during the resting and plateau phases of the heart cycle. This is probably due to the fact that rather accurate a priori information is available during these time intervals. The problem of computing the transmembrane potential at an arbitrary time instance during a heart beat is still an open problem.

Wavefront-based models for inverse electrocardiography

We introduce two wavefront-based methods for the inverse problem of electrocardiography, which we term wavefront-based curve reconstruction (WBCR) and wavefront-based potential reconstruction (WBPR). In the WBCR approach, the epicardial activation wavefront is modeled as a curve evolving on the heart surface, with the evolution governed by factors derived phenomenologically from prior measured data. The body surface potential/wavefront relationship is modeled via an intermediate mapping of wavefront to epicardial potentials, again derived phenomenologically. In the WBPR approach, we iteratively construct an estimate of epicardial potentials from an estimated wavefront curve according to a simplified model and use it as an initial solution in a Tikhonov regularization scheme. Initial simulation results using measured canine epicardial data show considerable improvement in reconstructing activation wavefronts and epicardial potentials with respect to standard Tikhonov solutions. In particular the WBCR method accurately finds the anisotropic propagation early after epicardial pacing, and the WBPR method finds the wavefront (regions of sharp gradient of the potential) both accurately and with minimal smoothing.

Use of group theory in the selection and description of regularization methods for functional source imaging

Commonly, functional source imaging problems are "partial" rather than "ordinary" inverse problems--wherein the defining operator consists of component operators that individually do not address all variables of the unknown. When this ordinary-to-partial transition is minimally constrained, algebraic principles can be used to derive a favored methodology--which we do here. The resulting Isotropy method is compared to two other regularization methods proposed for functional source imaging (Kalman and Joint Regularization). This theoretical support for the favored status of the Isotropy method is consistent with its favorable computational performance in low prior information settings, as indicated in recent publications.

Rapid construction of a patient-specific torso model from 3D ultrasound for non-invasive imaging of cardiac electrophysiology

One of the main limitations in using inverse methods for non-invasively imaging cardiac electrical activity in a clinical setting is the difficulty in readily obtaining high-quality data sets to reconstruct accurately a patient-specific geometric model of the heart and torso. This issue was addressed by investigation into the feasibility of using a pseudo-3D ultrasound system and a hand-held laser scanner to reconstruct such a model. This information was collected in under 20 min prior to a catheter ablation or pacemaker study in the electrophysiology laboratory. Using the models created from these data, different activation field maps were computed using several different inverse methods. These were independently validated by comparison of the earliest site of activation with the physical location of the pacing electrodes, as determined from orthogonal fluoroscopy images. With an estimated average geometric error of approximately 8 mm, it was also possible to reconstruct the site of initial activation to within 17.3 mm and obtain a quantitatively realistic activation sequence. The study demonstrates that it is possible rapidly to construct a geometric model that can then be used non-invasively to reconstruct an activation field map of the heart.